Such systolic inequalities are proved for surfaces such as non-flat dimensional torus \((\) its proof is not in the literature, but mentioned in [1]}\()\) ,
projective space [1]},
and higher genus ones [3]}, [4]}, [5]}.
For higher dimensional Riemannian manifolds,
Gromov showed the existence of the systolic constant \(C_d\)  for essential manifolds [5]}.
Here a manifold is said to be essential if the fundamental class \([M]\in H_d(M)\)  does not vanish via the natural homomorphism to the homology of the Eilenberg–Maclean space \(H_d(M)\rightarrow H_d(K(\pi _1M,1))\) .
This class contains closed surfaces,
closed aspherical manifolds and lens spaces.
